MinT - Best Known (3, 88, s)-Nets in Base 32

Best Known (3, 88, s)-Nets in Base 32

(3, 88, 64)-Net over F₃₂ — Constructive and digital

Digital (3, 88, 64)-net over F₃₂, using

net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
  - a function field by SÃ©mirat [i]

(3, 88, 186)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (3, 88, 187)-net over F₃₂, because

extracting embedded orthogonal array [i] would yield linear OA(32⁸⁸, 187, F₃₂, 85) (dual of [187, 99, 86]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(32⁸⁷, 91, S₃₂, 85), but
    - the linear programming bound shows that MÂ â‰¥ 3218 248804 644806 290772 103657 191613 826913 486259 137011 802066 676331 137713 410455 220883 164990 453062 699224 855954 368127 744971 436177 620882 948096 / 34443 > 32⁸⁷ [i]
  2. linear OA(32⁹⁹, 187, F₃₂, 96) (dual of [187, 88, 97]-code), but
    - discarding factors / shortening the dual code would yield linear OA(32⁹⁹, 132, F₃₂, 96) (dual of [132, 33, 97]-code), but
      - residual code [i] would yield OA(32³, 35, S₃₂, 3), but
        1 times truncation [i] would yield OA(32², 34, S₃₂, 2), but
        bound for OAs with strength kÂ =Â 2 [i]
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1055 > 32² [i]

(3, 88, 299)-Net in Base 32 — Upper bound on s

There is no (3, 88, 300)-net in base 32, because

2 times m-reduction [i] would yield (3, 86, 300)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32⁸⁶, 300, S₃₂, 83), but
  - the linear programming bound shows that MÂ â‰¥ 2371 082253 783878 425248 553856 940579 401884 068817 703494 133598 203004 557014 579464 478627 708630 292957 311115 811849 678611 509090 437162 872253 887040 890369 787102 756864 / 839361 482026 182080 269297 > 32⁸⁶ [i]